The Scaling Rules- Trimmer (1)

8/17/2019 The Scaling Rules- Trimmer (1)

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The Scaling of

Micromechanical Devices by

William Trimmer

This material is adapted from the article “Microrobots and Micromechanical Systems”

by W. S. N. Trimmer, Sensors and Actuators, Volume 1, Number !, September 1",

pa#es $%& ' $"&, and other sources. The boo( )Micromechanics and M*MS) has this

and other interestin# articles on small mechanical systems+ published by the ***

-ress, number -/!0, SN 0'&"0!'10"2'!. A more detailed analysis of the scalin#

of electroma#netic forces is #i3en in the Appendi4 to “Microrobots and

Micromechanical Systems.”

A nice description of scalin# is #i3en in “Trimmer5s Vertical rac(et Notation” in the boo( “6undamentals of Microfabrication” by Marc Madou, SN 0'"/!'/21'1,

7 -ress 1&.

To desi#n micromechanical actuators, it is helpful to understand ho8 forces scale. A

simple notation for understandin# multiple force la8s and e9uations is described

belo8. This notation is used to describe ho8 different forces scale into the small :and

lar#e; domain.

This paper uses a matri4 formalism to describe the scalin# la8s. This nomenclature

sho8s a number of different force la8s in a sin#le e9uation. n this notation, the si


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sho8s four cases for the force la8. The top force la8 scales as S, ne4t scales as S

s9uared or S$ : hope this appears as S superscript $ on your screen, one ne3er (no8s

8ith the 8eb;, the ne4t as S!, and the bottom as S/. The scalin# of the time re9uired

to mo3e an ob?ect usin# these forces is #i3en as

The top element in e9uation $ is S1.2. This is ho8 the time scales 8hen the force

scales as S1. The second element sho8s that t scales as S1 8hen the force scales as

S$. The third and forth element sho8 ho8 the time scales 8hen the force scales as S!

and S/ respecti3ely. This notation is used consistently throu#hout this paper. A dash

@B means that this case does not apply.

This 3ertical brac(et notation can be used for other scalin# la8s. 6or e4ample, if onehad a desire the top element could refer to the case 8here the force scales as S%. =r

the top element could represent to the case 8here the acceleration scales as S 1, and the

second element represent the case 8here the acceleration scales as S$ , ... . These

3ertical brac(ets can be defined for the con3enience of the problem at hand. All that

is needed is the initial definition of 8hat each element represents. :*9uation 1 is this

definition in our present case.;


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Magnetic Forces

This Section e4amines the scalin# of ma#netic forces caused by the interactions of

electrical currents. Three cases are e4aminedC A; constant temperature rise from thecenter to the e4terior of the coil 8indin#s, ; constant heat flo8 per unit surface

area of the coil 8indin#s, and ; constant electrical current density in the coil

8indin#s. Assumption A; leads to forces that scale as S$, assumption ; leads to

forces that scale as S!, and assumption ; leads to forces that scale as S/. These three

cases are depicted in e9uation $.2. The deri3ation of this force scalin# re9uires a bit

of math and 8ill not be #i3en here. This deri3ation is #i3en in the appendi4 of

“Microrobots and Micromechanical Systems.”

: ha3e no idea 8hy my e9uation editor #i3es rounded brac(ets in the e9uation abo3e,instead of the s9uare brac(ets 8ish. =h, 8ell.;

The abo3e force scalin# is for the case of t8o electrical currents interactin#. As S

decreases, these forces decrease because it is difficult to #enerate lar#e ma#netic fields

8ith small coils of 8ire :electroma#nets;. Do8e3er permanent ma#nets maintain

their stren#th as they are scaled do8n in si S0, and hence a

8ire 8ith one tenth the area carries one tenth the current. The heat #enerated per

3olume of 8indin#s is constant for this case. The force #enerated for this constant

current case scales as @ S/ B E S! F, i e., 8hen the system decreases by a factor of ten in

si


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factor of ten thousand. learly this is not a stron# micro force. :Do8e3er, on the

#alactic scale, ma#netic forces become truly impressi3e. Ioo(in# at the spiral arms of

the S and S #ala4ies, 8onder ho8 these lar#e ma#netic fields effect the comple4

t8istin# of #alactic matter.;

ase ; Since heat can be more easily conducted out of a small 3olume, it is possibleto run isolated small motors 8ith hi#her current densities than assumed abo3e.

Do8e3er, increasin# the current density ma(es the motors much less efficient. :Note,

electronics is usually much more 8asteful of po8er than the micromechanical

components, and the po8er used by the motor is often insi#nificant.; f the heat flo8

per unit surface area of the 8indin#s is constant, the current density in the 8ires scales

as G > S'0.2 This increase in current density for small systems increases the force

#enerated, and the force scales as @ S! B E S$.2 F .

ase A; A third possible constraint on the ma#netic system as it is scaled do8n is the

ma4imum temperature the 8ire and insulation can 8ithstand. f the system

parameters are scaled so that there is a constant temperature difference bet8een the

8indin#s and surroundin# en3ironment, then the current density scales as G > : S '1 ;

and the force scales as @ S$ B E S$ F As 8ill be discussed later, forces that scale as

: S$ ; are useful in small systems. Dence in many micro desi#ns, it may be

ad3anta#eous to use the a##ressi3e increase in current density assumed in case A.

n summary, the currents re9uired for the different force la8s scale asC

These current scalin# are the result of the assumptions in ase A, ase , and ase ;

and #enerate the forcesC


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n desi#nin# micro electroma#nets, one must also consider electromi#ration. At hi#h

current densities, tiny 8ires are deformed by the current and the 8ire can brea(. 6or

e4ample thin aluminum interconnects at current densities hi#her than 2 4 10 2 AH cm$

sho8 the de3elopment of 3oids and hilloc(s 8hich can lead to coil failure. The

temperature, composition and len#th of time the conductor is used ha3e a lar#e effect

on the electromi#ration. :7eferencesC @1B A.Scor


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0.2 , then the force has the e3en better scalin# of 6 > S1 . When the si


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8here S6 represents the scalin# of the force 6. Dere only the time to accelerate has

been calculated, but an e9ual time is needed to decelerate, and both these times scale

in the same 8ay. 6or the forces #i3en in e9uation :1;, the accelerations and transit

times can be e4pressed as

and


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*3en in the 8orst case, 8here 6 > S/ :the bottom element;, the time re9uired to

perform a tas( remains constant, t > S0 , 8hen the system is scaled do8n. Knder more

fa3orable force scalin#, for e4ample, the 6 > S$ scalin# case, the time re9uired

decreases as t > S1 8ith the scale. A system ten times smaller can perform an

operation ten times faster. This is an obser3ation that 8e (no8 intuiti3elyC smallthin#s tend to be 9uic(.

nertial forces tend to become insi#nificant in the small domain, and in many cases

(inematics may replace dynamics. This 8ill probably lead to interestin# ne8 control

strate#ies.

Power generated and dissipated

As the scale of a system is chan#ed, one 8ants to (no8 ho8 the po8er produced

depends upon the force la8s. 6or e4ample, consider the unit cube abo3e, 8hich is first

accelerated and then decelerated. The po8er, -, or the 8or( done on the ob?ect per

unit time is

The scalin# of each of the terms on the ri#ht is (no8n.


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The po8er that can be produced per unit 3olume : V> S! ; is

When the force scales as S$ then the po8er per unit 3olume scales as S'1 . 6or

e4ample, 8hen the scale decreases by a factor of ten, the po8er that can be #enerated

per unit 3olume increases by a factor of ten. 6or force la8s 8ith a hi#her po8er than

S$ , the po8er #enerated per 3olume de#rades as the scale si


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8hen desi#nin# small systems. :-lease remember, these force la8s depend upon

#eneral assumptions, there is al8ays room to be cle3er.;

6or the ma#netic case, one may be concerned about the po8er dissipated by the

resisti3e loss of the 8ires. The po8er due to this resisti3e loss, -7 , is

8here A is the cross section of the 8ire, :rho; is the resisti3ity of the 8ire, and I is the

len#th of the 8ire. This #i3es

8here :A I; is the 3olume. The resisti3ity scales as S0 and the 3olume scales as S!

and from e9uation :!; abo3e,

Dence the po8er dissipated scales asC


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and the po8er per unit 3olume isC

6or the ma#netic case A; 8here force scales as S$, the po8er that must be dissipated

per unit 3olume scales as S'$ , or, 8hen the scale is decreased by a factor of ten, a

hundred times as much po8er must be dissipated 8ithin a set 3olume. This ma#netic

case is bad if one is concerned about po8er density or the amount of coolin# needed.

f po8er dissipation or coolin# are not a critical concern, then this scalin# case

produces more substantial forces. n the future, superconductors may #i3e us stron#er

micro electroma#nets.

Summary of the scaling results

The force has been found to scale in one of four different 8aysC @ S1 B , @ S$ B , @ S! B ,

and @ S/ B . f the scale si


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and


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6or the force la8s that beha3e as @ S1 B or @ S$ B , the acceleration increases as one

scales do8n the system. The po8er that can be produced per unit 3olume alsoincreases for these t8o la8s. The surface tension scales ad3anta#eously, @ S1 B ,

ho8e3er, it is not clear ho8 to use this force in most applications. iolo#ical forces

also scale 8ell, @ S$ B but may be difficult to implement. *lectrostatic and pressure

related forces appear to be 9uite useful forces in the small domain.

The main page is http:www!trimmer!net"eturn to MEMS at trimmer!net #TM$

'

William Trimmers Dome -a#e Source for Micromechanics O M*MS O

=ther Web Sites O Gournals O oo(s.

' f you ha3e comments or su##estions, or an interest in usin# M*MS, e'mail

William Trimmer at 8illiamPtrimmer.net
http://home.earthlink.net/~trimmerw/mems/index.htmlmailto:[email protected]://home.earthlink.net/~trimmerw/mems/index.htmlmailto:[email protected]

The Scaling Rules- Trimmer (1)

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